Moreover, since the class map \(D^b_c(X) \rightarrow K_0(D^b_c(X))\)  is onto, \(\chi _{st}\)  is already an epimorphism on \(D^b_c(X)\) . The usual functors in sheaf theory, which respect the corresponding category of bounded constructible complexes, induce via \(\chi _{st}\)  well-defined group homomorphisms on the level of constructible functions (see, e.g., [1]}).
